The arithmetic sequence $a_i$ is defined by the formula: $a_1 = 566$ $a_i = a_{i - 1} -4$ Find the sum of the first $90$ terms in the sequence.
Solution: Getting started Let's write out the first few terms of the series: $566 + 562 + 558 + 554...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $4$ less than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {566})$ and the number of terms $(n = {90})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $90 -1= 89$ terms after the first term. The sequence decreases by $4$ for each new term. So, the sequence decreases by a total of $89 \cdot 4 = 356$ from where it starts at $566$. That means the last term must be $566-356 = {210}$. In other words, $a_n = {210}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{90}}&= \dfrac {\left({566} + {210} \right)}{2} \cdot {90} \\\\ S_{{90}} &= 388 \left(90\right) \\\\ S_{{90}} &= 34{,}920\end{aligned}$ The answer $ 34{,}920 $